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We present Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions based on the foundational ideas of function encoders. We decompose the task of learning operators into two parts: learning sets of basis functions for both the input and output spaces and learning a potentially nonlinear mapping between the coefficients of the basis functions. B2B operator learning circumvents many challenges of prior works, such as requiring data to be at fixed locations, by leveraging classic techniques such as least squares to compute the coefficients. It is especially potent for linear operators, where we compute a mapping between bases as a single matrix transformation with a closed-form solution. Furthermore, with minimal modifications and using the deep theoretical connections between function encoders and functional analysis, we derive operator learning algorithms that are directly analogous to eigen-decomposition and singular value decomposition. We empirically validate B2B operator learning on seven benchmark operator learning tasks and show that it demonstrates a two-orders-of-magnitude improvement in accuracy over existing approaches on several benchmark tasks. 

Developing autonomous agents that can strategize and cooperate with humans under information asymmetry is challenging without effective communication in natural language. We introduce a shared-control game, where two players collectively control a token in alternating turns to achieve a common objective under incomplete information. We formulate a policy synthesis problem for an autonomous agent in this game with a human as the other player. To solve this problem, we propose a communication-based approach comprising a language module and a planning module. The language module translates natural language messages into and from a finite set of flags, a compact representation defined to capture player intents. The planning module leverages these flags to compute a policy using an asymmetric information-set Monte Carlo tree search with flag exchange algorithm we present. We evaluate the effectiveness of this approach in a testbed based on Gnomes at Night, a search-and-find maze board game. Results of human subject experiments show that communication narrows the information gap between players and enhances human-agent cooperation efficiency with fewer turns.